Mathematics and the Internet: A Source of Enormous Confusion and Great Potential

Graph theory models the Internet mathematically, and a number of plausible mathematically intersecting network models for the Internet have been developed and studied. Simultaneously, Internet researchers have developed methodology to use real data to validate, or invalidate, proposed Internet models. The authors look at these parallel developments, particularly as they apply to scale-free network models of the preferential attachment type

Robustness of airline alliance route networks

The aim of this study is to analyze the robustness of the three major airline alliances’ (i.e., Star Alliance, oneworld and SkyTeam) route networks. Firstly, the normalization of a multi-scale measure of vulnerability is proposed in order to perform the analysis in networks with different sizes, i.e., number of nodes. An alternative node selection criterion is also proposed in order to study robustness and vulnerability of such complex networks, based on network efficiency. And lastly, a new procedure – the inverted adaptive strategy – is presented to sort the nodes in order to anticipate network breakdown. Finally, the robustness of the three alliance networks are analyzed with (1) a normalized multi-scale measure of vulnerability, (2) an adaptive strategy based on four different criteria and (3) an inverted adaptive strategy based on the efficiency criterion. The results show that Star Alliance has the most resilient route network, followed by SkyTeam and then oneworld. It was also shown that the inverted adaptive strategy based on the efficiency criterion – inverted efficiency – shows a great success in quickly breaking networks similar to that found with betweenness criterion but with even better results.Peer ReviewedPostprint (author’s final draft

Software systems through complex networks science: Review, analysis and applications

Complex software systems are among most sophisticated human-made systems, yet only little is known about the actual structure of 'good' software. We here study different software systems developed in Java from the perspective of network science. The study reveals that network theory can provide a prominent set of techniques for the exploratory analysis of large complex software system. We further identify several applications in software engineering, and propose different network-based quality indicators that address software design, efficiency, reusability, vulnerability, controllability and other. We also highlight various interesting findings, e.g., software systems are highly vulnerable to processes like bug propagation, however, they are not easily controllable

Dynamics of Tipping Cascades on Complex Networks

Tipping points occur in diverse systems in various disciplines such as ecology, climate science, economy or engineering. Tipping points are critical thresholds in system parameters or state variables at which a tiny perturbation can lead to a qualitative change of the system. Many systems with tipping points can be modeled as networks of coupled multistable subsystems, e.g. coupled patches of vegetation, connected lakes, interacting climate tipping elements or multiscale infrastructure systems. In such networks, tipping events in one subsystem are able to induce tipping cascades via domino effects. Here, we investigate the effects of network topology on the occurrence of such cascades. Numerical cascade simulations with a conceptual dynamical model for tipping points are conducted on Erd\H{o}s-R\'enyi, Watts-Strogatz and Barab\'asi-Albert networks. Additionally, we generate more realistic networks using data from moisture-recycling simulations of the Amazon rainforest and compare the results to those obtained for the model networks. We furthermore use a directed configuration model and a stochastic block model which preserve certain topological properties of the Amazon network to understand which of these properties are responsible for its increased vulnerability. We find that clustering and spatial organization increase the vulnerability of networks and can lead to tipping of the whole network. These results could be useful to evaluate which systems are vulnerable or robust due to their network topology and might help to design or manage systems accordingly.Comment: 22 pages, 12 figure

Centrality scaling in large networks

Betweenness centrality lies at the core of both transport and structural vulnerability properties of complex networks, however, it is computationally costly, and its measurement for networks with millions of nodes is near impossible. By introducing a multiscale decomposition of shortest paths, we show that the contributions to betweenness coming from geodesics not longer than L obey a characteristic scaling vs L, which can be used to predict the distribution of the full centralities. The method is also illustrated on a real-world social network of 5.5*10^6 nodes and 2.7*10^7 links

Multiscale vulnerability of complex networks

We present a novel approach to quantify the vulnerability of a complex network, i.e., the capacity of a graph to maintain its functional performance under random damages or malicious attacks. The proposed measure represents a multiscale evaluation of vulnerability, and makes use of combined powers of the links' betweenness. We show that the proposed approach is able to properly describe some cases for which earlier measures of vulnerability fail. The relevant applications of our method for technological network design are outlined